Lie algebroid
E581260
A Lie algebroid is a geometric structure that generalizes Lie algebras and tangent bundles, encoding infinitesimal symmetries on manifolds via a vector bundle with a Lie bracket and an anchor map.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
geometric structure
ⓘ
mathematical structure ⓘ |
| admits | cohomology theory ⓘ |
| anchorMapTarget | tangent bundle of the base manifold ⓘ |
| anchorMapType | vector bundle morphism ⓘ |
| appearsIn |
generalized geometry
ⓘ
integrability problems of Lie groupoids ⓘ |
| associatedWith | Lie groupoid ⓘ |
| bracketProperty |
Leibniz rule with respect to anchor
ⓘ
antisymmetric ⓘ satisfies Jacobi identity ⓘ |
| bracketType | R-bilinear map on sections ⓘ |
| cohomologyUsedFor |
characteristic classes
ⓘ
deformation theory ⓘ |
| compatibilityCondition | anchor map is Lie algebra homomorphism on sections ⓘ |
| definedOn | smooth manifold ⓘ |
| encodes |
infinitesimal symmetries
ⓘ
infinitesimal transformations on manifolds ⓘ |
| field |
Lie theory
ⓘ
Poisson geometry ⓘ differential geometry ⓘ |
| generalizes |
Lie algebra
ⓘ
tangent bundle ⓘ |
| hasAnchorMap | bundle map from the vector bundle to the tangent bundle ⓘ |
| hasBase | smooth manifold ⓘ |
| hasBracket | Lie bracket on space of sections ⓘ |
| hasComponent |
Lie bracket
ⓘ
anchor map ⓘ vector bundle ⓘ |
| hasDualObject | Lie groupoid ⓘ |
| hasGeneralization |
VB-algebroid
ⓘ
higher Lie algebroid ⓘ |
| hasMorphisms | Lie algebroid morphisms ⓘ |
| hasTotalSpace | vector bundle over a manifold ⓘ |
| introducedBy | Jean Pradines NERFINISHED ⓘ |
| isInfinitesimalVersionOf | Lie groupoid ⓘ |
| morphismType | vector bundle maps compatible with brackets and anchors ⓘ |
| relatedConcept |
Courant algebroid
NERFINISHED
ⓘ
Lie-Rinehart algebra NERFINISHED ⓘ Poisson manifold ⓘ |
| specialCase |
Lie algebra as Lie algebroid over a point
ⓘ
tangent bundle with usual Lie bracket of vector fields ⓘ |
| studiedIn | global analysis on manifolds ⓘ |
| usedIn |
Poisson geometry
NERFINISHED
ⓘ
deformation quantization ⓘ foliation theory ⓘ gauge theory ⓘ nonlinear connection theory ⓘ |
| yearIntroduced | 1967 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.